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Lecture 10
Categorical Logic
Categorical statements
Why do we need Categorical Logic?
Propositional logic does not cover all valid
logical forms.
It is an important part of logic, but not the only part
of logic.
Is the following argument valid?
All human beings are mortal;
Socrates is a human being;
So, Socrates is mortal
This is a valid argument. Can it be explained by
propositional logic? Not quite.
Argument:
All human beings are mortal;
Socrates is a human being;
So, Socrates is mortal
1.
2.
3.
Analysis (Modus Ponens?):
a)
b)
c)
If Socrates is a human being, then Socrates is mortal.
Socrates is a human being;
So, Socrates is mortal
But statement 1 and statement (a) are not quite the
same thing.
Another argument
Some four-legged
creatures are gnus.
All gnus are herbivores.
Therefore, some fourlegged creatures are
herbivores.
Is this argument valid?
Argument:
Some four-legged creatures are gnus.
All gnus are herbivores.
Therefore, some four-legged creatures are
herbivores.
This argument is valid, but it cannot be
explained by propositional logic.
We need some kind of new logic.
Categorical Logic
Categorical logic was
discovered long long ago
by Aristotle (384-322
BCE).
The arguments we have
discussed are called
syllogism.
Aristotle discovered all
valid forms of syllogism.
Categorical Logic
Categorical logic studies inference between categorical
statements.
It is the focus of today’s lecture to explain the structure of
categorical statements.
Typical inferences between categorical statements:
Syllogism:
Latin Square:
Case: All As are Bs, All Bs are Cs, so all As are Cs.
Case: All As are Bs; therefore it is not the case that some As are not
Bs.
All As are Bs; so Some As are Bs.
Conversion:
Case: All As are Bs; so All Bs are As.
Categorical Statements
Statements in categorical logic have a specific
structure.
All human beings are mortal.
Some four-legged creatures are gnus.
They have the following structure:
quantifier + subject phrase + <link verb ‘be’> +
predicate phrase
All categorical statements are structured like this.
The link verb ‘be’ is called ‘copula’.
Quantification
Categorical statements are different from
statements in propositional logic: they have
quantifiers to quantify a statement.
Consider:
All human beings are mortal.
Some people are rich.
No pigs are able to fly.
Here, all, some, and no are quantifiers.
Subject phrase and Predicate phrase
Subject phrase and predicate phrase are
understood as classes, i.e. a set of objects that fall
in the phrase .
Human being: it is a class/set of all objects that are
human beings.
Mortal: it is a class/set of all objects that are mortal.
Socrates: this is an one-man class, which contains
only one person-Socrates.
Ordinary language
Sometimes the predicate phrases are not
explicitly referring to a class; but we can always
transform them to its corresponding class.
Sometimes copula is not present either, and we can
add one.
Examples:
I love apples. == I am the one who loves apples
Bees are angry == Bees are one of the things that
are angry.
Four Kinds of Categorical
Statements
All S are P.
It asserts that every member of class S is a member of
class P.
Some S are P.
At least one member of S is also in the class P.
Note the use of the word ‘Some’: it does not imply that
there are more than one member of S is in P.
No S are P.
No member of S is in the class P.
Some S are not P.
At least one member of S is not in the class P.
Traditional Terminology
In traditional logic, each of four kinds of
categorical statements is given a names.
A: All S are P. (Universal Affirmative)
E: No S are P. (Universal Negative)
I: Some S are P. (Particular Affirmative)
O: Some S are not P. (Particular Negative)
Further explanations
There are two ways to characterize a categorical
statement:
Quality: whether the statement confirms (a confirmative) or
negates (a negative);
Quantity: whether it has a universal (all) or a particular
quantifier (some).
Types of Categorical Statements
A: all As are Bs: it is a Universal Confirmative.
E: no As are Bs: it is the same as “All As are not Bs”; so it is
a Universal Negative.
I and O statements can be similarly understood.
Translations
In ordinary language, categorical statements are
often not put in a standard form. So translations
are often needed in order for us to have a
precise and accurate understanding of these
statements.
The standard form is also important for us to
capture the argument relation between categorical
statements.
Missing Copula or Quantifier
Copula or quantifiers are not present in these
statements:
Dogs love meat.
Workers should get paid.
Solution: add the missing parts (without
changing the meanings of the statements)
All dogs are the animals that love meat.
All workers are the people who should get paid.
Terms without Nouns
Some statements may not have nouns as
predicates:
Roses are red;
All ducks swim.
Solution: replace them with a noun phrase or a
noun clause
All roses are red things.
All ducks are animals that swim.
Singular Statements
What about statements with a singular subject term?
G. W. Bush is a good president.
George loves Starbucks.
Rule: treat singular statement as A-statement.
Singular term is an one-object class. Then it says about
everything in the class.
G. W. Bush is a good president == all people who are
identical with G. W. Bush is a good president.
George loves Starbucks==all people who is identical with
George love Starbucks.
Other expressions for the Universal
Quantifier
Cases
Every soldier is a warrior.
Each one of you is responsible.
Whoever is a doctor earns a lot of money.
Any tiger can be dangerous.
These are all A-statements; these quantifiers are
the same as ‘all’.
Every soldier is a warrior.
All soldiers are warriors.
E-Statement
Cases:
Nobody loves Ray.
Nothing is better than pure love.
None of the animals are alive.
Rule: these are all E-statements. Treat these
quantifiers as ‘no’.
Nobody loves Ray.
No people are the people who love Ray.
I-Statement
Cases
Most students are honest.
Many people are retiring late.
A few dogs got killed.
At least one person is missing.
Almost all the cats have four legs.
There are government employees who are spies.
Rule: these are all I-statements. Treat all these
quantifiers as ‘some’.
There is a significant difference between ‘most’ ‘many’ and
‘some’, but it is beyond our concern here.
O-Statements
Cases:
Not all the rich people are smart.
There are government employees who are not
qualified.
Some rich people are not smart.
Some government employees are not qualified.
Most Democrats are not in favor of the war.
Some Democrats are not in favor of the war
Other Structures
Not all A are B ≠ No A are B.
Not all A are B:
This is an O-Statement: some As are not Bs.
No A are B
This is an E-Statements. No As are Bs.
Case:
Not all Republicans support the war.
No Republicans support the war.
‘Only’ & ‘only if’
‘Only’?
‘only if’?
Only rich people are invited.
Only if one studies logic one gets smart.
Translation rule:
These statements are A-statements. The term after ‘only’ or
‘only if’ is predicate term.
Only rich people are invited == all invited people are rich
ones.
Only if one studies logic one gets smart. == all smart people
are those who study logic.
‘The Only’
Case:
Translation rule:
The only guests invited are boys.
Cockroaches are the only survivors.
These are A-statements; the term that occurs after ‘the only’
is the subject term.
Translation:
The only guests invited are boys == all guests invited are
boys.
Cockroaches are the only survivors == all survivors are
cockroaches.
Try these exercises in the book!
Page 263, Ex. 7.2: Questions No. 4, 6, 8, 10, 14,
15.
Page 263-4, Ex. 7.3: No. 10, 13, 14, 17.
Ex. 7.2
#4 People who whisper lie.
All people who whisper are people who lie.
#6 Only if something has a back beat is it a rock-androll song.
All rock-and-roll songs are things with a back beat.
A-statement
A-statement (note the predicate and the subject)
Only, only if: what follows are predicates of an A-statement
The only: what follows are subjects of an A-statement
#8: Nothing that is a snake is a mammal.
No snakes are mammals. E-statement.
#10: The only good human is a dead human.
All good humans are dead humans.
#14: There is no excellence without difficulty.
No excellence is without difficulty.
E-statement.
Or: All excellent things are things with difficulty.
A-statement.
A-statement
#15: Jonathan is not a very brave pilot.
No people who are identical to Jonathan are very brave pilots.
E-statement.
Is this an A-statement? Not really.
A statement vs. E-statement
A statement: All S are P.
E-statement: No S are P.
What about?
All S are not P:
No S are not P:
E-statement
A-statement
Why? Venn diagrams show they are so.
Compare:
Some S are P.
Some S are not P.
Ex. 7.3: No. 10, 13, 14, 17
10: People who love only once in their lives are
shallow people. (Oscar Wilde)
All the people who love only once in their lives
are shallow people.
A-statement
13: Many socialists are not communists.
Some socialists are not communists.
O-statement
14: All prejudices may be traced to the intestines.
All prejudices are things that may be traced to the
intestines.
17: He that is born to be hanged shall never be
drowned.
All people who are born to be hanged are people who
shall never be drowned.
A-statement.
A -statement
No people who are born to be hanged are the people
who shall be drowned
E-statement
Quiz
In this quiz you will be give a set of categorical
statements in English, and you are asked to
translated them into standard form of a
categorical statement, and indicate its type (A, E,
I, O).
It is similar to the exercises we did in the previous
slides.